After the discretisation in
space equation (REF ) becomes an \(N+1\) -dimensional system of
ordinary differential equations (ODEs)
in \(t\)  of the form
\(\tilde{v}_{t}=f(\tilde{v})\)  which can be numerically integrated in
time with standard techniques. The discussion in
[1]} shows that Chebyshev differentiation matrices have a
conditioning of the order \(\mbox{cond}(D^{3})=O(N^{6})\) . Thus explicit time
integration schemes are problematic since stability conditions would
necessitate prohibitively small time steps.
Therefore, here we apply an implicit method, and since we are
interested in capturing rapid oscillations in the expected DSWs, we use a fourth order method.
